nLab
Berkovich space

Idea

Berkovich analytic spaces are a version of analytic spaces over nonarchimedean fields. Unlike the rigid analytic spaces (see rigid analytic geometry) of Tate, which are locally defined via maximal spectra of Tate algebras flued via rigidified Grothendieck topology, the Berkovich analytic spaces are defined with respect to the analytic spectrum, due to Vladimir Berkovich; this spectrum can be viewed consisting of the data of prime ideal plus the extension of the norm to the residue field; thus the Berkovich spectrum has far more points (though less than, say, Huber’s adic spaces which may also contain valuations of higher order).

For more background see analytic geometry.

Definition of Berkovich analytic spaces

Let $k$ be a non-archimedean field.

Definition

Given $n\in \mathbb{N}$ and positive elements $\{r_1,\cdots ,r_n\in k\}$, consider the sub-power series algebra over $k$ defined by

$$\{\frac{1}{r_1}{T}_1,\cdots ,\frac{1}{r_n}{T}_n\}:=\{\sum _{\nu }{a}_{\nu }{T}^{\nu }\mid \underset{\mid \nu \mid \to 0}{\lim }\mid a_{\nu }\mid r^{\nu }=0\}.$$\{\frac{1}{r_1} T_1 , \cdots, \frac{1}{r_n}T_n\} := \left\{ \sum_\nu a_\nu T^\nu | \lim_{{\vert \nu\vert} \to 0} {\vert a_\nu \vert} r^\nu = 0 \right\} \,.

This is a commutative Banach algebra over $k$ with norm $\parallel f\parallel =\max \mid a_{\nu }\mid r^{\nu }$.

A $k$-affinoid algebra is a commutative Banach $k$-algebra $A$ for which there exists $n$ and $\{r_i\}$ as above and an epimorphism

$$\{\frac{1}{r_1}{T}_1,\cdots ,\frac{1}{r_n}{T}_n\}\to A$$\{\frac{1}{r_1} T_1 , \cdots, \frac{1}{r_n}T_n\} \to A

such that the norm on $A$ is the quotient norm.

If one can choose here $r_i=1$ for all $i$ then $A$ is called strictly $k$-affinoid.

The category of $k$-affinoid spaces is the opposite category of the category of $k$-affinoid algebras and bounded homomorphisms between them.

Via the analytic spectrum ${\mathrm{Spec}}_{\mathrm{an}}$ there is a topological space associated with any $k$-affinoid space. Often this underlying topological space is referred to as the analytic space.

Definition

An affinoid domain in an affoinoid space $X={\mathrm{Spec}}_{\mathrm{an}}A$ is a closed subset $V\subset X$ such that there is a homomorphism of $k$-affinoid spaces

$$\varphi :{\mathrm{Spec}}_{\mathrm{an}}{A}_V\to X$$\phi : Spec_{an} A_V \to X

for some $A_V$, whose image is $V$, and such that every other morphism of $k$-affinoid spaces into $X$ whose image is contained in $V$ uniquely factors through this morphism.

Definition

A $k$-analytic space is a topological space equipped with an atlas by affinoid domains.

See Berkovich 98, paragraph 2.

Properties

Local contractibility

A complex analytic manifold and a smooth complex analytic space is locally isomorphic to a polydisk and hence is trivially a locally contractible space. But over a non-archimedean field analytic spaces no longer need to be locally isomorphic to polydisks (but $p$-adic polydisks are still contractible (Berkovich 90)). The following result establishes, under mild conditions, that general analytic spaces are nevertheless locally contractible.

Assume that the valuation on the ground field $k$ is nontrivial.

Definition

A $k$-analytic space $X$ is called locally embeddable in a smooth space if each point of $X$ has an open neighbourhood isomorphic to a strictly $k$-analytic domain in smooth $k$-analytic space.

Theorem

Every $k$-analytic space which is locally embeddable in a smooth space, def. 4, is a locally contractible space.

More precisely, every point of a locally smooth $k$-analytic space has an open neighbourhood $U$ which is contractible, and which is a union $U={\cup }_{i=1}^{\mathrm{\infty }}{U}_i$ of analytic domains.

The local contractibility is Berkovich (1999), theorem 9.1. The refined statment in terms of inductive systems of analytic domains is in Berkovich (2004).

Examples

• analytic affine line

Applications

• The proof of the local Langlands conjecture for ${\mathrm{GL}}_n$ by Harris–Taylor uses étale cohomology on non-archimedean analytic spaces (in the sense of Berkovich) to construct the required Galois representations over local fields.

Related concepts

• non-commutative analytic space

References

Introductions and reviews

Basic notions are listed in

• M. Temkin, Non-archimedean analytic spaces (pdf slides)

A review of basic definitions and facts about affinoid and rigid $k$-analytic spaces can be found in

• Gaëtan Chenevier, lecture 5 (pdf)

See also the references at rigid analytic geometry.

A review of definitions and results on $k$-analytic spaces is in

• Vladimir Berkovich, $p$-Adic analytic spaces, in Proceedings of the International Congress of Mathematicians, Berlin, August 1998, Doc. Math. J. DMV, Extra Volume ICM II (1998), 141-151 (pdf)

A more detailed set of lecture notes along these lines is

• Vladimir Berkovich, Non-archimedean analytic spaces, lectures at the Advanced School on $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

The Berkovich spectrum is discussed around section 2.1.4 in the course notes

• Frédéric Paugam, Global analytic geometry and the functional equation (2010) (pdf)

A exposition of examples of Berkovits spectra is in

• Scott Carnahan, Berkovich spaces I (web)

Original articles

• Vladimir Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, (1990) 169 pp.

• Vladimir Berkovich, Étale cohomology for non-Archimedean analytic spaces, Publ. Math. IHES 78 (1993), 5-161.

Discussion of local contractibility of smooth $k$-analytic spaces is in

• Vladimir Berkovich, Smooth $p$-adic analytic spaces are locally contractible, Invent. Math. 137 1-84 (1999) (pdf)

• Vladimir Berkovich, Smooth p-adic analytic spaces are locally contractible. II, in Geometric Aspects of Dwork Theory, Walter de Gruyter & Co., Berlin, (2004), 293-370. (pdf)

Relation to other topics

On the relation to buildings:

• Annette Werner, Buildings and Berkovich Spaces (pdf)

Relation to integration theory

• Vladimir Berkovich, Integration of 1-forms on $p$-adic analytic spaces, Princeton University Press,

Aspects of the homotopy theory/étale homotopy of analytic spaces are discussed in

• Aise Johan de Jong, Étale fundamental groups of non-archimedean analytic spaces, Mathematica, 97 no. 1-2 (1995), p. 89-118 (numdam)